Yong Chen - Industrial Data Analytics for Diagnosis and Prognosis

For a data set y i, i = 1, 2,…, n obtained by multiplying each xi by a constant a , i.e., yi = axi , i = 1, 2,…, n , it is easy to see that

The sample variance measures the spread of the data and is defined as

(2.1)

The square root of the sample variance, s = √ s 2, is called the sample standard deviation . The sample standard deviation is of the same measurement unit as the observations. For yi = axi , i = 1,2,…, n , its sample variance is

If each of the n observations of a data set is measured on two variables x 1and x 2, let ( x 11, x 21,..., x n 1) and ( x 12, x 22,..., x n 2) denote the n observations on x 1and x 2, respectively. The sample covariance of x 1and x 2is defined as

(2.2)

where x̄ 1and x̄ 2are the sample means of x 1and x 2, respectively. The value of sample covariance of two variables is affected by the linear association between them. From ( 2.2), if x 1and x 2have a strong positive linear association, they are usually both above their means or both below their means. Consequently, the product ( x i1− x ¯ 1)( x i2− x ¯ 2) will typically be positive and their sample covariance will have a large positive value. On the other hand, if x 1and x 2have a strong negative linear association, the product ( x i1− x ¯ 1)( x i2− x ¯ 2) will typically be negative and their sample covariance will have a negative value. If y 1and y 2are obtained by multiplying each measurement of x 1and x 2with a 1and a 2, respectively, it is easy to see from ( 2.2) that the sample covariance of y 1and y 2is

(2.3)

Equation ( 2.3) says that if the measurements are scaled, for example by changing measurement units, the sample covariance will be scaled correspondingly. The sample covariance’s dependence on the measurement units makes it difficult to determine how large a sample covariance indicates a strong (linear) association between two variables. The sample correlation defined as follows is a measure of linear association that does not depend on the measurement units, or scaling of the variables

(2.4)

where s 1and s 2are the sample standard deviation of x 1and x 2, respectively. The sample correlation ranges between −1 and 1, with values close to 1, −1, and 0 indicating a strong positive linear association, a strong negative linear association, and no linear association, respectively.

Example 2.2To illustrate the calculation of summary statistics, we take a random sample of 10 observations, as shown in Table 2.1, from the auto.specdata set on the variables curb.weight, length, and width. We use x i, i =1,2,3, to represent the three variables:

Table 2.1 A random sample of 10 observations from the auto. spec data set.

To obtain the sample covariance for the variables curb.weightand lengthin the data set in Table 2.1, we first calculate the sample means x̄ 1, x̄ 2, and as:

By ( 2.2), the sample covariance of the two variables can be obtained as

The s 12value of 4316.8 itself cannot tell us whether the two variables have a strong or weak (linear) relationship. Such information can be provided by the correlation. To evaluate the sample correlation, we first need the sample variance of x 1and x 2. By ( 2.1), we have

By ( 2.4), we have

which is close to 1 and corresponding to a strong positive linear association between the curb weight and length of cars.

Example 2.3In R, the sample mean, variance, covariance, and correlation can be found using functions mean(), var(), cov(), and cor(), respectively. For example, the following Rcodes can be used to find the sample mean and sample variance of curb.weight, and the sample covariance and correlation between curb.weightand length, in the auto.specdata set.